Factors and Multiples

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Factors and Multiples
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Factors of a Number

• A number may be made by multiplying two or more
  other numbers together. The numbers that are
  multiplied together are called factors of the
  final number.

Factors of 24
1 X 24 2 X 12 3 X 8 4 X 6
Multiplication combinations to reach 24
1, 2, 3, 4, 6, 8, 12, 24 A list of its factors
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Greatest Common Factor (GCF)

• Greatest common factor is the greatest whole
  number that is a factor of two or more given
  whole numbers.

Example
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What is the greatest common factor of 36 and 48?

• Factors of 36

1, 2, 3, 4, 6, 9, 12, 18, 36
36
1
2
18
3
12
4
9
6
Factors of 48
1, 2, 3, 4, 6, 8,12, 16, 24, 48
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What is the greatest common factor of 36 and 48?
Factors of 36
1, 2, 3, 4, 6, 9, 12, 18, 36
12
6
4
3
2
1
Factors of 48
1, 2, 3, 4, 6, 8,12, 16, 24, 48
12
6
4
3
2
1
Which factors are common to both numbers?
Which is the greatest common factor?
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Greatest Common Factor (GCF)

• That was too much work. What happened if we
  forgot to include the 12 in one of the factor
  list?

Want to learn a different way thats easier?
Duh!
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Birthday Cake Method to find GCF
Think of any number that divides evenly into both
36 and 48 (a factor of both)
Divide both numbers by 2.
For this example, Ill use 2.
18 24
36 48
2
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Birthday Cake Method to find GCF
Think of any number that divides evenly into both
18 and 24 (a factor of both)
Divide both numbers by 3.
6
8
For this example, Ill use 3.
3
18 24
36 48
2
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Birthday Cake Method to find GCF
Continue this process until there is no other
number besides 1 that is a factor of both numbers.
3 and 4 have no common factor except 1... so you
can stop dividing.
3
4
6
8
2
3
18 24
36 48
2
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Birthday Cake Method to find GCF
The product of the numbers along the side of the
cake is the GCF of 36 and 48.
2 x 3 x 2 12
3
4

6
8
2
3
18 24
36 48
2
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Birthday Cake Method to find GCF
Observe another example. Find the GCF of 64 and
80.
8 x 2 16
4
5
no common factor except 1
8
32 40
64 80
2
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Multiples of a Number
The products of a number with the natural numbers
1, 2, 3, 4, 5, ... are called the multiples of
the number.
7 x 1 7 7 x 2 14 7 x 3 21 7 x 4 28
For example
So, the multiples of 7 are 7, 14, 21, 28, and so
on.
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Multiples of a Number

• Further examples
• multiples of 2 are 2, 4, 6, 8,
• multiples of 3 are 3, 6, 9, 12,
• multiples of 4 are 4, 8, 12, 16,

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Common Multiples

• Multiples that are common to two or more numbers
  are said to be common multiples. For example
• Multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16,
  18,
• Multiples of 3 are 3, 6, 9, 12, 15, 18,
• So, common multiples of 2 and 3 are 6, 12, 18,

6
12
18
12
18
6
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Example 2 Find the common multiples of 4 and
6. Solution Multiples of 4 are 4, 8, 12, 16, 20,
24, 28, 32, 36, Multiples of 6 are 6, 12, 18,
24, 30, 36, So, the common multiples of 4 and 6
are 12, 24, 36,
24
12
36
36
24
12
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Least Common Multiple (LCM)

• The smallest common multiple of two or more
  numbers is called the least common multiple
  (LCM).

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Least Common Multiple (LCM)
To find the least common multiple (LCM) of two or
more numbers, list the multiples of the larger
number and stop when you find a multiple of the
other number.  This is the LCM.
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Least Common Multiple (LCM)
Find the least common multiple of 6 and
9. Solution List the multiples of 9 and stop
when you find a multiple of 6. Multiples of 9
are 9, 18, Multiples of 6 are 6, 12, 18,
                      
LCM is 18
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Birthday Cake Method to find LCM
With the birthday cake method, you can find both
the GCF and LCM at the same time!
Lets use a previous example
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Birthday Cake Method to find LCM
Find the least common multiple of 36 and 48.
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Birthday Cake Method to find LCM
Find the product of the numbers going up and on
top of the cake.
Lets make the birthday cake!
144 144 is the LCM of 36 and 48
3
4
x
x
6
8
2
x
3
18 24
x
36 48
2
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Birthday Cake Method to find LCM
Another example Find the LCM of 16 and 9.
Do 16 and 9 have any common factors besides 1?
No, they dont. So their LCM is their product
16 9
?
X
144
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Birthday Cake Method to find LCM
Another example Find the LCM of 5, 6 and 8.
Finding the answer by listing multiples of the
largest number (8) would eventually lead you to
the answer of 120.
Multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64,
72, 80, 88, 96, 104, 112, 120, Stop at 120 as
it is a multiple of both 5 and 6.
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Birthday Cake Method to find LCM
The birthday cake method can be used to find the
LCM of 5, 6 and 8. Watch!
3 4
5
x
x
120
x
5 6 8
2